Introduction to Mathematical Physics/Dual of a vectorial space

Dual of a vectorial space

Definition

When $E$ has a finite dimension, then $E^{*}$ has also a finite dimension and its dimension is equal to the dimension of $E$. If $E$ has an infinite dimension, $E^{*}$ has also an infinite dimension but the two spaces are not isomorphic.

Template:IMP/label

Tensors

In this appendix, we introduce the fundamental notion of tensor\index{tensor} in physics. More information can be found in ([#References|references]) for instance. Let $E$ be a finite dimension vectorial space. Let $e_i$ be a basis of $E$. A vector $X$ of $E$ can be referenced by its components $x^i$ is the basis $e_i$: Template:IMP/eq In this chapter the repeated index convention (or {\bf Einstein summing convention}) will be used. It consists in considering that a product of two quantities with the same index correspond to a sum over this index. For instance: Template:IMP/eq or Template:IMP/eq To the vectorial space $E$ corresponds a space $E^{*}$ called the dual of $E$. A element of $E^{*}$ is a linear form on $E$: it is a linear mapping $p$ that maps any vector $Y$ of $E$ to a real. $p$ is defined by a set of number $x_i$ because the most general form of a linear form on $E$ is: Template:IMP/eq A basis $e^i$ of $E^{*}$ can be defined by the following linear form Template:IMP/eq where ${\delta }_i^j$ is one if $i=j$ and zero if not. Thus to each vector $X$ of $E$ of components $x^i$ can be associated a dual vector in $E^{*}$ of components $x_i$: Template:IMP/eq The quantity Template:IMP/eq is an invariant. It is independent on the basis chosen. On another hand, the expression of the components of vector $X$ depend on the basis chosen. If ${\omega }_k^i$ defines a transformation that maps basis $e_i$ to another basis $e{\text{'}}_i$ Template:IMP/label Template:IMP/eq we have the following relation between components $x_i$ of $X$ in $e_i$ and $x{\text{'}}_i$ of $X$ in $e{\text{'}}_i$: Template:IMP/label Template:IMP/eq This comes from the identification of Template:IMP/eq and Template:IMP/eq Equations eqcov and eqcontra define two types of variables: covariant variables that are transformed like the vector basis. $x_i$ are such variables. Contravariant variables that are transformed like the components of a vector on this basis. Using a physicist vocabulary $x_i$ is called a covariant vector and $x^i$ a contravariant vector.

Covariant and contravariant components of a vector $X$.} Template:IMP/label

Let $x_i$ and $y_j$ two vectors of two vectorial spaces $E_1$ and $E_2$. The tensorial product space $E_1\otimes E_2$ is the vectorial space such that there exist a unique isomorphism between the space of the bilinear forms of $E_1\times E_2$ and the linear forms of $E_1\otimes E_2$. A bilinear form of $E_1\times E_2$ is: Template:IMP/eq It can be considered as a linear form of $E_1\otimes E_2$ using application $\otimes$ from $E_1\times E_2$ to $E_1\otimes E_2$ that is linear and distributive with respect to $+$. If $e_i$ is a basis of $E_1$ and $f_j$ a basis of $E_2$, then Template:IMP/eq $e_i\otimes e_j$ is a basis of $E_1\otimes E_2$. Thus tensor $x_i{y}_j=T_{ij}$ is an element of $E_1\otimes E_2$. A second order covariant tensor is thus an element of $E^{*}\otimes E^{*}$. In a change of basis, its components $aij$ are transformed according the following relation: Template:IMP/eq Now we can define a tensor on any rank of any variance. For instance a tensor of third order two times covariant and one time contravariant is an element $a$ of $E^{*}\otimes E^{*}\otimes E$ and noted $a_{ij}^k$.

A second order tensor is called symmetric if $a_{ij}=a_{ji}$. It is called antisymmetric is $a_{ij}=-a_{ji}$.

Pseudo tensors are transformed slightly differently from ordinary tensors. For instance a second order covariant pseudo tensor is transformed according to: Template:IMP/eq where $det(\omega )$ is the determinant of transformation $\omega$.

Let us introduce two particular tensors.

• The Kronecker symbol ${\delta }_{ij}$ is defined by: Template:IMP/eq It is the only second order tensor invariant in $R^3$ by rotations.
• The signature of permutations tensor $e_{ijk}$ is defined by:

It is the only pseudo tensor of rank 3 invariant by rotations in $R^3$. It verifies the equality: Template:IMP/eq

Let us introduce two tensor operations: scalar product, vectorial product.

• Scalar product $a.b$ is the contraction of vectors $a$ and $b$ : Template:IMP/eq
• vectorial product of two vectors $a$ and $b$ is: Template:IMP/eq

From those definitions, following formulas can be showed:

$\begin{array}{ccc}a.(b\wedge c)& =& a_i(b\wedge c{)}_i\\ & =& a_i{ϵ}_{ijk}{b}_j{c}_k\\ & =& {ϵ}_{ijk}{a}_i{b}_j{c}_k\\ & =& \left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|\end{array}$

Here is useful formula: Template:IMP/eq

Green's theorem

Green's theorem allows one to transform a volume calculation integral into a surface calculation integral. Template:IMP/thm Here are some important Green's formulas obtained by applying Green's theorem: Template:IMP/eq Template:IMP/eq Template:IMP/eq